Maksimov and Kolovsky, Equation (4)

Percentage Accurate: 86.2% → 99.9%

Time: 11.0s

Alternatives: 14

Speedup: 1.4×

• 48.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Initial Program: 86.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))

C

double code(double J, double l, double K, double U) {
	return ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = ((j * (exp(l) - exp(-l))) * cos((k / 2.0d0))) + u
end function

Java

public static double code(double J, double l, double K, double U) {
	return ((J * (Math.exp(l) - Math.exp(-l))) * Math.cos((K / 2.0))) + U;
}

Python

def code(J, l, K, U):
	return ((J * (math.exp(l) - math.exp(-l))) * math.cos((K / 2.0))) + U

Julia

function code(J, l, K, U)
	return Float64(Float64(Float64(J * Float64(exp(l) - exp(Float64(-l)))) * cos(Float64(K / 2.0))) + U)
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = ((J * (exp(l) - exp(-l))) * cos((K / 2.0))) + U;
end

Wolfram

code[J_, l_, K_, U_] := N[(N[(N[(J * N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_0 \leq -0.04 \lor \neg \left(t\_0 \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;\left(t\_0 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (- (exp l) (exp (- l)))))
   (if (or (<= t_0 -0.04) (not (<= t_0 6e-5)))
     (+ (* (* t_0 J) (cos (/ K 2.0))) U)
     (+
      U
      (*
       l
       (* (cos (* K 0.5)) (* J (fma 0.3333333333333333 (pow l 2.0) 2.0))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = exp(l) - exp(-l);
	double tmp;
	if ((t_0 <= -0.04) || !(t_0 <= 6e-5)) {
		tmp = ((t_0 * J) * cos((K / 2.0))) + U;
	} else {
		tmp = U + (l * (cos((K * 0.5)) * (J * fma(0.3333333333333333, pow(l, 2.0), 2.0))));
	}
	return tmp;
}

Julia

function code(J, l, K, U)
	t_0 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_0 <= -0.04) || !(t_0 <= 6e-5))
		tmp = Float64(Float64(Float64(t_0 * J) * cos(Float64(K / 2.0))) + U);
	else
		tmp = Float64(U + Float64(l * Float64(cos(Float64(K * 0.5)) * Float64(J * fma(0.3333333333333333, (l ^ 2.0), 2.0)))));
	end
	return tmp
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.04], N[Not[LessEqual[t$95$0, 6e-5]], $MachinePrecision]], N[(N[(N[(t$95$0 * J), $MachinePrecision] * N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(l * N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] * N[(J * N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.0400000000000000008 or 6.00000000000000015e-5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   if -0.0400000000000000008 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 6.00000000000000015e-5

   1. Initial program 74.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \color{blue}{\ell \cdot \left(0.3333333333333333 \cdot \left(J \cdot \left({\ell}^{2} \cdot \cos \left(0.5 \cdot K\right)\right)\right) + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   4. Step-by-step derivation

      1. associate-*r* 99.9%

         \[\leadsto \ell \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(\left(J \cdot {\ell}^{2}\right) \cdot \cos \left(0.5 \cdot K\right)\right)} + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      2. associate-*r* 99.9%

         \[\leadsto \ell \cdot \left(\color{blue}{\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \cos \left(0.5 \cdot K\right)} + 2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) + U \]

      3. associate-*r* 99.9%

         \[\leadsto \ell \cdot \left(\left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right)\right) \cdot \cos \left(0.5 \cdot K\right) + \color{blue}{\left(2 \cdot J\right) \cdot \cos \left(0.5 \cdot K\right)}\right) + U \]

      4. distribute-rgt-out 99.9%

         \[\leadsto \ell \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \left(0.3333333333333333 \cdot \left(J \cdot {\ell}^{2}\right) + 2 \cdot J\right)\right)} + U \]

      5. *-commutative 99.9%

         \[\leadsto \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\color{blue}{\left(J \cdot {\ell}^{2}\right) \cdot 0.3333333333333333} + 2 \cdot J\right)\right) + U \]

      6. associate-*l* 99.9%

         \[\leadsto \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(\color{blue}{J \cdot \left({\ell}^{2} \cdot 0.3333333333333333\right)} + 2 \cdot J\right)\right) + U \]

      7. *-commutative 99.9%

         \[\leadsto \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\left(0.3333333333333333 \cdot {\ell}^{2}\right)} + 2 \cdot J\right)\right) + U \]

      8. *-commutative 99.9%

         \[\leadsto \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{2}\right) + \color{blue}{J \cdot 2}\right)\right) + U \]

      9. distribute-lft-out 99.9%

         \[\leadsto \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \color{blue}{\left(J \cdot \left(0.3333333333333333 \cdot {\ell}^{2} + 2\right)\right)}\right) + U \]

      10. fma-define 99.9%

          \[\leadsto \ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \color{blue}{\mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)}\right)\right) + U \]

   5. Simplified 99.9%

      \[\leadsto \color{blue}{\ell \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.04 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(\cos \left(K \cdot 0.5\right) \cdot \left(J \cdot \mathsf{fma}\left(0.3333333333333333, {\ell}^{2}, 2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ t_1 := e^{\ell} - e^{-\ell}\\ \mathbf{if}\;t\_1 \leq -0.04 \lor \neg \left(t\_1 \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;\left(t\_1 \cdot J\right) \cdot t\_0 + U\\ \mathbf{else}:\\ \;\;\;\;U + t\_0 \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))) (t_1 (- (exp l) (exp (- l)))))
   (if (or (<= t_1 -0.04) (not (<= t_1 6e-5)))
     (+ (* (* t_1 J) t_0) U)
     (+ U (* t_0 (* J (* l (+ 2.0 (* 0.3333333333333333 (pow l 2.0))))))))))

C

double code(double J, double l, double K, double U) {
	double t_0 = cos((K / 2.0));
	double t_1 = exp(l) - exp(-l);
	double tmp;
	if ((t_1 <= -0.04) || !(t_1 <= 6e-5)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * pow(l, 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((k / 2.0d0))
    t_1 = exp(l) - exp(-l)
    if ((t_1 <= (-0.04d0)) .or. (.not. (t_1 <= 6d-5))) then
        tmp = ((t_1 * j) * t_0) + u
    else
        tmp = u + (t_0 * (j * (l * (2.0d0 + (0.3333333333333333d0 * (l ** 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	double t_1 = Math.exp(l) - Math.exp(-l);
	double tmp;
	if ((t_1 <= -0.04) || !(t_1 <= 6e-5)) {
		tmp = ((t_1 * J) * t_0) + U;
	} else {
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * Math.pow(l, 2.0))))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	t_0 = math.cos((K / 2.0))
	t_1 = math.exp(l) - math.exp(-l)
	tmp = 0
	if (t_1 <= -0.04) or not (t_1 <= 6e-5):
		tmp = ((t_1 * J) * t_0) + U
	else:
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * math.pow(l, 2.0))))))
	return tmp

Julia

function code(J, l, K, U)
	t_0 = cos(Float64(K / 2.0))
	t_1 = Float64(exp(l) - exp(Float64(-l)))
	tmp = 0.0
	if ((t_1 <= -0.04) || !(t_1 <= 6e-5))
		tmp = Float64(Float64(Float64(t_1 * J) * t_0) + U);
	else
		tmp = Float64(U + Float64(t_0 * Float64(J * Float64(l * Float64(2.0 + Float64(0.3333333333333333 * (l ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	t_0 = cos((K / 2.0));
	t_1 = exp(l) - exp(-l);
	tmp = 0.0;
	if ((t_1 <= -0.04) || ~((t_1 <= 6e-5)))
		tmp = ((t_1 * J) * t_0) + U;
	else
		tmp = U + (t_0 * (J * (l * (2.0 + (0.3333333333333333 * (l ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[l], $MachinePrecision] - N[Exp[(-l)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.04], N[Not[LessEqual[t$95$1, 6e-5]], $MachinePrecision]], N[(N[(N[(t$95$1 * J), $MachinePrecision] * t$95$0), $MachinePrecision] + U), $MachinePrecision], N[(U + N[(t$95$0 * N[(J * N[(l * N[(2.0 + N[(0.3333333333333333 * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < -0.0400000000000000008 or 6.00000000000000015e-5 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l)))

   1. Initial program 99.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   if -0.0400000000000000008 < (-.f64 (exp.f64 l) (exp.f64 (neg.f64 l))) < 6.00000000000000015e-5

   1. Initial program 74.6%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\ell} - e^{-\ell} \leq -0.04 \lor \neg \left(e^{\ell} - e^{-\ell} \leq 6 \cdot 10^{-5}\right):\\ \;\;\;\;\left(\left(e^{\ell} - e^{-\ell}\right) \cdot J\right) \cdot \cos \left(\frac{K}{2}\right) + U\\ \mathbf{else}:\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.3333333333333333 \cdot {\ell}^{2}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \lor \neg \left(\ell \leq 4\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -4.1) (not (<= l 4.0)))
   (+ U (* (cos (/ K 2.0)) (* 0.0003968253968253968 (* J (pow l 7.0)))))
   (+ U (* l (* 2.0 (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.1) || !(l <= 4.0)) {
		tmp = U + (cos((K / 2.0)) * (0.0003968253968253968 * (J * pow(l, 7.0))));
	} else {
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-4.1d0)) .or. (.not. (l <= 4.0d0))) then
        tmp = u + (cos((k / 2.0d0)) * (0.0003968253968253968d0 * (j * (l ** 7.0d0))))
    else
        tmp = u + (l * (2.0d0 * (j * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -4.1) || !(l <= 4.0)) {
		tmp = U + (Math.cos((K / 2.0)) * (0.0003968253968253968 * (J * Math.pow(l, 7.0))));
	} else {
		tmp = U + (l * (2.0 * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -4.1) or not (l <= 4.0):
		tmp = U + (math.cos((K / 2.0)) * (0.0003968253968253968 * (J * math.pow(l, 7.0))))
	else:
		tmp = U + (l * (2.0 * (J * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -4.1) || !(l <= 4.0))
		tmp = Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(0.0003968253968253968 * Float64(J * (l ^ 7.0)))));
	else
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -4.1) || ~((l <= 4.0)))
		tmp = U + (cos((K / 2.0)) * (0.0003968253968253968 * (J * (l ^ 7.0))));
	else
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -4.1], N[Not[LessEqual[l, 4.0]], $MachinePrecision]], N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.0003968253968253968 * N[(J * N[Power[l, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -4.0999999999999996 or 4 < l

   1. Initial program 99.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 87.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 87.3%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 87.3%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 87.3%

      \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   if -4.0999999999999996 < l < 4

   1. Initial program 75.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 99.9%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 99.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Taylor expanded in l around 0 99.0%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{6}\right) + 2 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Taylor expanded in l around 0 99.1%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   9. Step-by-step derivation

      1. *-commutative 99.1%

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)}\right) + U \]

      2. *-commutative 99.1%

         \[\leadsto 2 \cdot \left(J \cdot \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot \ell\right)\right) + U \]

      3. associate-*r* 99.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \ell\right)} + U \]

      4. *-commutative 99.1%

         \[\leadsto 2 \cdot \left(\left(J \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) \cdot \ell\right) + U \]

      5. associate-*l* 99.1%

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot \ell} + U \]

      6. *-commutative 99.1%

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   10. Simplified 99.1%

       \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -4.1 \lor \neg \left(\ell \leq 4\right):\\ \;\;\;\;U + \cos \left(\frac{K}{2}\right) \cdot \left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{7}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U \cdot \left(1 + \frac{\cos \left(K \cdot 0.5\right)}{U} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (* U (+ 1.0 (* (/ (cos (* K 0.5)) U) (* J (* l 2.0)))))
   (+ U (* J (* l (+ 2.0 (* 0.0003968253968253968 (pow l 6.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = U * (1.0 + ((cos((K * 0.5)) / U) * (J * (l * 2.0))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * pow(l, 6.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.1d0)) then
        tmp = u * (1.0d0 + ((cos((k * 0.5d0)) / u) * (j * (l * 2.0d0))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.0003968253968253968d0 * (l ** 6.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.1) {
		tmp = U * (1.0 + ((Math.cos((K * 0.5)) / U) * (J * (l * 2.0))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * Math.pow(l, 6.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.1:
		tmp = U * (1.0 + ((math.cos((K * 0.5)) / U) * (J * (l * 2.0))))
	else:
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * math.pow(l, 6.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = Float64(U * Float64(1.0 + Float64(Float64(cos(Float64(K * 0.5)) / U) * Float64(J * Float64(l * 2.0)))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.0003968253968253968 * (l ^ 6.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.1)
		tmp = U * (1.0 + ((cos((K * 0.5)) / U) * (J * (l * 2.0))));
	else
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * (l ^ 6.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(U * N[(1.0 + N[(N[(N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision] / U), $MachinePrecision] * N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.0003968253968253968 * N[Power[l, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.10000000000000001

   1. Initial program 79.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 73.4%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 73.4%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot J\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*r* 73.4%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 73.4%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in U around inf 76.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 76.2%

         \[\leadsto U \cdot \left(1 + \color{blue}{\frac{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}{U}}\right) \]

      2. associate-*r* 76.2%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}}{U}\right) \]

      3. *-commutative 76.2%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(J \cdot 2\right)} \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right) \]

   8. Simplified 76.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{\left(J \cdot 2\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* 76.2%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(\left(J \cdot 2\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)}}{U}\right) \]

      2. *-un-lft-identity 76.2%

         \[\leadsto U \cdot \left(1 + \frac{\left(\left(J \cdot 2\right) \cdot \ell\right) \cdot \cos \left(0.5 \cdot K\right)}{\color{blue}{1 \cdot U}}\right) \]

      3. times-frac 76.2%

         \[\leadsto U \cdot \left(1 + \color{blue}{\frac{\left(J \cdot 2\right) \cdot \ell}{1} \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}}\right) \]

      4. *-commutative 76.2%

         \[\leadsto U \cdot \left(1 + \frac{\left(J \cdot 2\right) \cdot \ell}{1} \cdot \frac{\cos \color{blue}{\left(K \cdot 0.5\right)}}{U}\right) \]

   10. Applied egg-rr 76.2%

       \[\leadsto U \cdot \left(1 + \color{blue}{\frac{\left(J \cdot 2\right) \cdot \ell}{1} \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}}\right) \]
   11. Step-by-step derivation

       1. /-rgt-identity 76.2%

          \[\leadsto U \cdot \left(1 + \color{blue}{\left(\left(J \cdot 2\right) \cdot \ell\right)} \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}\right) \]

       2. associate-*l* 76.2%

          \[\leadsto U \cdot \left(1 + \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right)} \cdot \frac{\cos \left(K \cdot 0.5\right)}{U}\right) \]

       3. *-commutative 76.2%

          \[\leadsto U \cdot \left(1 + \left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \frac{\cos \color{blue}{\left(0.5 \cdot K\right)}}{U}\right) \]

   12. Simplified 76.2%

       \[\leadsto U \cdot \left(1 + \color{blue}{\left(J \cdot \left(2 \cdot \ell\right)\right) \cdot \frac{\cos \left(0.5 \cdot K\right)}{U}}\right) \]

   if -0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 94.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 94.6%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 94.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 93.9%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Taylor expanded in K around 0 90.6%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U \cdot \left(1 + \frac{\cos \left(K \cdot 0.5\right)}{U} \cdot \left(J \cdot \left(\ell \cdot 2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + U \cdot \left(\left(J \cdot 2\right) \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (+ U (* U (* (* J 2.0) (/ (* l (cos (* K 0.5))) U))))
   (+ U (* J (* l (+ 2.0 (* 0.0003968253968253968 (pow l 6.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = U + (U * ((J * 2.0) * ((l * cos((K * 0.5))) / U)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * pow(l, 6.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.1d0)) then
        tmp = u + (u * ((j * 2.0d0) * ((l * cos((k * 0.5d0))) / u)))
    else
        tmp = u + (j * (l * (2.0d0 + (0.0003968253968253968d0 * (l ** 6.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.1) {
		tmp = U + (U * ((J * 2.0) * ((l * Math.cos((K * 0.5))) / U)));
	} else {
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * Math.pow(l, 6.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.1:
		tmp = U + (U * ((J * 2.0) * ((l * math.cos((K * 0.5))) / U)))
	else:
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * math.pow(l, 6.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = Float64(U + Float64(U * Float64(Float64(J * 2.0) * Float64(Float64(l * cos(Float64(K * 0.5))) / U))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.0003968253968253968 * (l ^ 6.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.1)
		tmp = U + (U * ((J * 2.0) * ((l * cos((K * 0.5))) / U)));
	else
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * (l ^ 6.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(U + N[(U * N[(N[(J * 2.0), $MachinePrecision] * N[(N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.0003968253968253968 * N[Power[l, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.10000000000000001

   1. Initial program 79.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 73.4%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 73.4%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot J\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*r* 73.4%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 73.4%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in U around inf 76.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   7. Step-by-step derivation

      1. associate-*r/ 76.2%

         \[\leadsto U \cdot \left(1 + \color{blue}{\frac{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}{U}}\right) \]

      2. associate-*r* 76.2%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}}{U}\right) \]

      3. *-commutative 76.2%

         \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(J \cdot 2\right)} \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right) \]

   8. Simplified 76.2%

      \[\leadsto \color{blue}{U \cdot \left(1 + \frac{\left(J \cdot 2\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
   9. Step-by-step derivation

      1. distribute-rgt-in 76.2%

         \[\leadsto \color{blue}{1 \cdot U + \frac{\left(J \cdot 2\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U} \cdot U} \]

      2. *-un-lft-identity 76.2%

         \[\leadsto \color{blue}{U} + \frac{\left(J \cdot 2\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U} \cdot U \]

      3. associate-/l* 80.9%

         \[\leadsto U + \color{blue}{\left(\left(J \cdot 2\right) \cdot \frac{\ell \cdot \cos \left(0.5 \cdot K\right)}{U}\right)} \cdot U \]

      4. *-commutative 80.9%

         \[\leadsto U + \left(\left(J \cdot 2\right) \cdot \frac{\ell \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}}{U}\right) \cdot U \]

   10. Applied egg-rr 80.9%

       \[\leadsto \color{blue}{U + \left(\left(J \cdot 2\right) \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right) \cdot U} \]

   if -0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 94.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 94.6%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 94.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 93.9%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Taylor expanded in K around 0 90.6%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + U \cdot \left(\left(J \cdot 2\right) \cdot \frac{\ell \cdot \cos \left(K \cdot 0.5\right)}{U}\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (<= (cos (/ K 2.0)) -0.1)
   (+ U (* l (* 2.0 (* J (cos (* K 0.5))))))
   (+ U (* J (* l (+ 2.0 (* 0.0003968253968253968 (pow l 6.0))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if (cos((K / 2.0)) <= -0.1) {
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * pow(l, 6.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if (cos((k / 2.0d0)) <= (-0.1d0)) then
        tmp = u + (l * (2.0d0 * (j * cos((k * 0.5d0)))))
    else
        tmp = u + (j * (l * (2.0d0 + (0.0003968253968253968d0 * (l ** 6.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if (Math.cos((K / 2.0)) <= -0.1) {
		tmp = U + (l * (2.0 * (J * Math.cos((K * 0.5)))));
	} else {
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * Math.pow(l, 6.0)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if math.cos((K / 2.0)) <= -0.1:
		tmp = U + (l * (2.0 * (J * math.cos((K * 0.5)))))
	else:
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * math.pow(l, 6.0)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if (cos(Float64(K / 2.0)) <= -0.1)
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(K * 0.5))))));
	else
		tmp = Float64(U + Float64(J * Float64(l * Float64(2.0 + Float64(0.0003968253968253968 * (l ^ 6.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if (cos((K / 2.0)) <= -0.1)
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	else
		tmp = U + (J * (l * (2.0 + (0.0003968253968253968 * (l ^ 6.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[LessEqual[N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision], -0.1], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * N[(2.0 + N[(0.0003968253968253968 * N[Power[l, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (/.f64 K #s(literal 2 binary64))) < -0.10000000000000001

   1. Initial program 79.3%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 95.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 95.0%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 95.0%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 95.0%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Taylor expanded in l around 0 91.9%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{6}\right) + 2 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Taylor expanded in l around 0 73.4%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   9. Step-by-step derivation

      1. *-commutative 73.4%

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)}\right) + U \]

      2. *-commutative 73.4%

         \[\leadsto 2 \cdot \left(J \cdot \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot \ell\right)\right) + U \]

      3. associate-*r* 73.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \ell\right)} + U \]

      4. *-commutative 73.4%

         \[\leadsto 2 \cdot \left(\left(J \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) \cdot \ell\right) + U \]

      5. associate-*l* 73.4%

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot \ell} + U \]

      6. *-commutative 73.4%

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   10. Simplified 73.4%

       \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   if -0.10000000000000001 < (cos.f64 (/.f64 K #s(literal 2 binary64)))

   1. Initial program 86.7%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 94.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 94.6%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 94.6%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 93.9%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Taylor expanded in K around 0 90.6%

      \[\leadsto \color{blue}{J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\frac{K}{2}\right) \leq -0.1:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 94.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)\right) \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (+
  U
  (*
   (cos (/ K 2.0))
   (* J (* l (+ 2.0 (* 0.0003968253968253968 (pow l 6.0))))))))

C

double code(double J, double l, double K, double U) {
	return U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.0003968253968253968 * pow(l, 6.0))))));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (cos((k / 2.0d0)) * (j * (l * (2.0d0 + (0.0003968253968253968d0 * (l ** 6.0d0))))))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (Math.cos((K / 2.0)) * (J * (l * (2.0 + (0.0003968253968253968 * Math.pow(l, 6.0))))));
}

Python

def code(J, l, K, U):
	return U + (math.cos((K / 2.0)) * (J * (l * (2.0 + (0.0003968253968253968 * math.pow(l, 6.0))))))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(cos(Float64(K / 2.0)) * Float64(J * Float64(l * Float64(2.0 + Float64(0.0003968253968253968 * (l ^ 6.0)))))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (cos((K / 2.0)) * (J * (l * (2.0 + (0.0003968253968253968 * (l ^ 6.0))))));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision] * N[(J * N[(l * N[(2.0 + N[(0.0003968253968253968 * N[Power[l, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 94.7%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 94.7%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Simplified 94.7%

   \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

6. Taylor expanded in l around inf 94.2%

   \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

7. Final simplification 94.2%

   \[\leadsto U + \cos \left(\frac{K}{2}\right) \cdot \left(J \cdot \left(\ell \cdot \left(2 + 0.0003968253968253968 \cdot {\ell}^{6}\right)\right)\right) \]
8. Add Preprocessing

Alternative 8: 83.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+48} \lor \neg \left(\ell \leq 9.5\right):\\ \;\;\;\;U + {\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.5e+48) (not (<= l 9.5)))
   (+ U (* (pow l 7.0) (* J 0.0003968253968253968)))
   (+ U (* 2.0 (* J (* l (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.5e+48) || !(l <= 9.5)) {
		tmp = U + (pow(l, 7.0) * (J * 0.0003968253968253968));
	} else {
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.5d+48)) .or. (.not. (l <= 9.5d0))) then
        tmp = u + ((l ** 7.0d0) * (j * 0.0003968253968253968d0))
    else
        tmp = u + (2.0d0 * (j * (l * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.5e+48) || !(l <= 9.5)) {
		tmp = U + (Math.pow(l, 7.0) * (J * 0.0003968253968253968));
	} else {
		tmp = U + (2.0 * (J * (l * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.5e+48) or not (l <= 9.5):
		tmp = U + (math.pow(l, 7.0) * (J * 0.0003968253968253968))
	else:
		tmp = U + (2.0 * (J * (l * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.5e+48) || !(l <= 9.5))
		tmp = Float64(U + Float64((l ^ 7.0) * Float64(J * 0.0003968253968253968)));
	else
		tmp = Float64(U + Float64(2.0 * Float64(J * Float64(l * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.5e+48) || ~((l <= 9.5)))
		tmp = U + ((l ^ 7.0) * (J * 0.0003968253968253968));
	else
		tmp = U + (2.0 * (J * (l * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.5e+48], N[Not[LessEqual[l, 9.5]], $MachinePrecision]], N[(U + N[(N[Power[l, 7.0], $MachinePrecision] * N[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(2.0 * N[(J * N[(l * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.49999999999999987e48 or 9.5 < l

   1. Initial program 99.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 93.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 93.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 93.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 93.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Taylor expanded in l around inf 93.1%

      \[\leadsto \color{blue}{0.0003968253968253968 \cdot \left(J \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   8. Step-by-step derivation

      1. associate-*r* 93.1%

         \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot J\right) \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 93.1%

         \[\leadsto \color{blue}{\left(J \cdot 0.0003968253968253968\right)} \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   9. Simplified 93.1%

      \[\leadsto \color{blue}{\left(J \cdot 0.0003968253968253968\right) \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   10. Taylor expanded in K around 0 73.6%

       \[\leadsto \left(J \cdot 0.0003968253968253968\right) \cdot \color{blue}{{\ell}^{7}} + U \]

   if -2.49999999999999987e48 < l < 9.5

   1. Initial program 76.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 94.3%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+48} \lor \neg \left(\ell \leq 9.5\right):\\ \;\;\;\;U + {\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\\ \mathbf{else}:\\ \;\;\;\;U + 2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+48} \lor \neg \left(\ell \leq 960\right):\\ \;\;\;\;U + {\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.5e+48) (not (<= l 960.0)))
   (+ U (* (pow l 7.0) (* J 0.0003968253968253968)))
   (+ U (* l (* 2.0 (* J (cos (* K 0.5))))))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.5e+48) || !(l <= 960.0)) {
		tmp = U + (pow(l, 7.0) * (J * 0.0003968253968253968));
	} else {
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.5d+48)) .or. (.not. (l <= 960.0d0))) then
        tmp = u + ((l ** 7.0d0) * (j * 0.0003968253968253968d0))
    else
        tmp = u + (l * (2.0d0 * (j * cos((k * 0.5d0)))))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.5e+48) || !(l <= 960.0)) {
		tmp = U + (Math.pow(l, 7.0) * (J * 0.0003968253968253968));
	} else {
		tmp = U + (l * (2.0 * (J * Math.cos((K * 0.5)))));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.5e+48) or not (l <= 960.0):
		tmp = U + (math.pow(l, 7.0) * (J * 0.0003968253968253968))
	else:
		tmp = U + (l * (2.0 * (J * math.cos((K * 0.5)))))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.5e+48) || !(l <= 960.0))
		tmp = Float64(U + Float64((l ^ 7.0) * Float64(J * 0.0003968253968253968)));
	else
		tmp = Float64(U + Float64(l * Float64(2.0 * Float64(J * cos(Float64(K * 0.5))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.5e+48) || ~((l <= 960.0)))
		tmp = U + ((l ^ 7.0) * (J * 0.0003968253968253968));
	else
		tmp = U + (l * (2.0 * (J * cos((K * 0.5)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.5e+48], N[Not[LessEqual[l, 960.0]], $MachinePrecision]], N[(U + N[(N[Power[l, 7.0], $MachinePrecision] * N[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(l * N[(2.0 * N[(J * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.49999999999999987e48 or 960 < l

   1. Initial program 99.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 93.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 93.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 93.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 93.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Taylor expanded in l around inf 93.1%

      \[\leadsto \color{blue}{0.0003968253968253968 \cdot \left(J \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   8. Step-by-step derivation

      1. associate-*r* 93.1%

         \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot J\right) \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 93.1%

         \[\leadsto \color{blue}{\left(J \cdot 0.0003968253968253968\right)} \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   9. Simplified 93.1%

      \[\leadsto \color{blue}{\left(J \cdot 0.0003968253968253968\right) \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   10. Taylor expanded in K around 0 73.6%

       \[\leadsto \left(J \cdot 0.0003968253968253968\right) \cdot \color{blue}{{\ell}^{7}} + U \]

   if -2.49999999999999987e48 < l < 960

   1. Initial program 76.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 95.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 95.7%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 95.7%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 94.9%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Taylor expanded in l around 0 94.3%

      \[\leadsto \color{blue}{\left(\ell \cdot \left(0.0003968253968253968 \cdot \left(J \cdot {\ell}^{6}\right) + 2 \cdot J\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   8. Taylor expanded in l around 0 94.3%

      \[\leadsto \color{blue}{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   9. Step-by-step derivation

      1. *-commutative 94.3%

         \[\leadsto 2 \cdot \left(J \cdot \color{blue}{\left(\cos \left(0.5 \cdot K\right) \cdot \ell\right)}\right) + U \]

      2. *-commutative 94.3%

         \[\leadsto 2 \cdot \left(J \cdot \left(\cos \color{blue}{\left(K \cdot 0.5\right)} \cdot \ell\right)\right) + U \]

      3. associate-*r* 94.3%

         \[\leadsto 2 \cdot \color{blue}{\left(\left(J \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \ell\right)} + U \]

      4. *-commutative 94.3%

         \[\leadsto 2 \cdot \left(\left(J \cdot \cos \color{blue}{\left(0.5 \cdot K\right)}\right) \cdot \ell\right) + U \]

      5. associate-*l* 94.3%

         \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right) \cdot \ell} + U \]

      6. *-commutative 94.3%

         \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]

   10. Simplified 94.3%

       \[\leadsto \color{blue}{\ell \cdot \left(2 \cdot \left(J \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+48} \lor \neg \left(\ell \leq 960\right):\\ \;\;\;\;U + {\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\\ \mathbf{else}:\\ \;\;\;\;U + \ell \cdot \left(2 \cdot \left(J \cdot \cos \left(K \cdot 0.5\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+48} \lor \neg \left(\ell \leq 4\right):\\ \;\;\;\;U + {\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -2.5e+48) (not (<= l 4.0)))
   (+ U (* (pow l 7.0) (* J 0.0003968253968253968)))
   (+ U (* J (* l 2.0)))))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.5e+48) || !(l <= 4.0)) {
		tmp = U + (pow(l, 7.0) * (J * 0.0003968253968253968));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-2.5d+48)) .or. (.not. (l <= 4.0d0))) then
        tmp = u + ((l ** 7.0d0) * (j * 0.0003968253968253968d0))
    else
        tmp = u + (j * (l * 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -2.5e+48) || !(l <= 4.0)) {
		tmp = U + (Math.pow(l, 7.0) * (J * 0.0003968253968253968));
	} else {
		tmp = U + (J * (l * 2.0));
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -2.5e+48) or not (l <= 4.0):
		tmp = U + (math.pow(l, 7.0) * (J * 0.0003968253968253968))
	else:
		tmp = U + (J * (l * 2.0))
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -2.5e+48) || !(l <= 4.0))
		tmp = Float64(U + Float64((l ^ 7.0) * Float64(J * 0.0003968253968253968)));
	else
		tmp = Float64(U + Float64(J * Float64(l * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -2.5e+48) || ~((l <= 4.0)))
		tmp = U + ((l ^ 7.0) * (J * 0.0003968253968253968));
	else
		tmp = U + (J * (l * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -2.5e+48], N[Not[LessEqual[l, 4.0]], $MachinePrecision]], N[(U + N[(N[Power[l, 7.0], $MachinePrecision] * N[(J * 0.0003968253968253968), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -2.49999999999999987e48 or 4 < l

   1. Initial program 99.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 93.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + 0.0003968253968253968 \cdot {\ell}^{2}\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 93.1%

         \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + \color{blue}{{\ell}^{2} \cdot 0.0003968253968253968}\right)\right)\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 93.1%

      \[\leadsto \left(J \cdot \color{blue}{\left(\ell \cdot \left(2 + {\ell}^{2} \cdot \left(0.3333333333333333 + {\ell}^{2} \cdot \left(0.016666666666666666 + {\ell}^{2} \cdot 0.0003968253968253968\right)\right)\right)\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in l around inf 93.1%

      \[\leadsto \left(J \cdot \left(\ell \cdot \left(2 + \color{blue}{0.0003968253968253968 \cdot {\ell}^{6}}\right)\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   7. Taylor expanded in l around inf 93.1%

      \[\leadsto \color{blue}{0.0003968253968253968 \cdot \left(J \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)\right)} + U \]
   8. Step-by-step derivation

      1. associate-*r* 93.1%

         \[\leadsto \color{blue}{\left(0.0003968253968253968 \cdot J\right) \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

      2. *-commutative 93.1%

         \[\leadsto \color{blue}{\left(J \cdot 0.0003968253968253968\right)} \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right) + U \]

   9. Simplified 93.1%

      \[\leadsto \color{blue}{\left(J \cdot 0.0003968253968253968\right) \cdot \left({\ell}^{7} \cdot \cos \left(0.5 \cdot K\right)\right)} + U \]

   10. Taylor expanded in K around 0 73.6%

       \[\leadsto \left(J \cdot 0.0003968253968253968\right) \cdot \color{blue}{{\ell}^{7}} + U \]

   if -2.49999999999999987e48 < l < 4

   1. Initial program 76.2%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   3. Taylor expanded in l around 0 94.2%

      \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
   4. Step-by-step derivation

      1. *-commutative 94.2%

         \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot J\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

      2. associate-*r* 94.2%

         \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Simplified 94.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

   6. Taylor expanded in K around 0 82.1%

      \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
   7. Step-by-step derivation

      1. *-commutative 82.1%

         \[\leadsto U + \color{blue}{\left(J \cdot \ell\right) \cdot 2} \]

      2. associate-*r* 82.1%

         \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

   8. Simplified 82.1%

      \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot 2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+48} \lor \neg \left(\ell \leq 4\right):\\ \;\;\;\;U + {\ell}^{7} \cdot \left(J \cdot 0.0003968253968253968\right)\\ \mathbf{else}:\\ \;\;\;\;U + J \cdot \left(\ell \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.7% accurate, 20.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq -3100000000 \lor \neg \left(\ell \leq 2.8 \cdot 10^{+95}\right):\\ \;\;\;\;J \cdot \frac{U}{J}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \end{array} \]

FPCore

(FPCore (J l K U)
 :precision binary64
 (if (or (<= l -3100000000.0) (not (<= l 2.8e+95))) (* J (/ U J)) U))

C

double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3100000000.0) || !(l <= 2.8e+95)) {
		tmp = J * (U / J);
	} else {
		tmp = U;
	}
	return tmp;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: tmp
    if ((l <= (-3100000000.0d0)) .or. (.not. (l <= 2.8d+95))) then
        tmp = j * (u / j)
    else
        tmp = u
    end if
    code = tmp
end function

Java

public static double code(double J, double l, double K, double U) {
	double tmp;
	if ((l <= -3100000000.0) || !(l <= 2.8e+95)) {
		tmp = J * (U / J);
	} else {
		tmp = U;
	}
	return tmp;
}

Python

def code(J, l, K, U):
	tmp = 0
	if (l <= -3100000000.0) or not (l <= 2.8e+95):
		tmp = J * (U / J)
	else:
		tmp = U
	return tmp

Julia

function code(J, l, K, U)
	tmp = 0.0
	if ((l <= -3100000000.0) || !(l <= 2.8e+95))
		tmp = Float64(J * Float64(U / J));
	else
		tmp = U;
	end
	return tmp
end

MATLAB

function tmp_2 = code(J, l, K, U)
	tmp = 0.0;
	if ((l <= -3100000000.0) || ~((l <= 2.8e+95)))
		tmp = J * (U / J);
	else
		tmp = U;
	end
	tmp_2 = tmp;
end

Wolfram

code[J_, l_, K_, U_] := If[Or[LessEqual[l, -3100000000.0], N[Not[LessEqual[l, 2.8e+95]], $MachinePrecision]], N[(J * N[(U / J), $MachinePrecision]), $MachinePrecision], U]

Derivation

1. Split input into 2 regimes

2. if l < -3.1e9 or 2.7999999999999998e95 < l

   1. Initial program 100.0%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 2.9%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in J around inf 18.8%

      \[\leadsto \color{blue}{J \cdot \left(8 \cdot \cos \left(0.5 \cdot K\right) + \frac{U}{J}\right)} \]

   6. Taylor expanded in U around inf 18.5%

      \[\leadsto J \cdot \color{blue}{\frac{U}{J}} \]

   if -3.1e9 < l < 2.7999999999999998e95

   1. Initial program 77.1%

      \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
   2. Add Preprocessing

   4. Applied egg-rr 45.8%

      \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   5. Taylor expanded in J around 0 65.0%

      \[\leadsto \color{blue}{U} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -3100000000 \lor \neg \left(\ell \leq 2.8 \cdot 10^{+95}\right):\\ \;\;\;\;J \cdot \frac{U}{J}\\ \mathbf{else}:\\ \;\;\;\;U\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.5% accurate, 28.4× speedup

Math

\[\begin{array}{l} \\ U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (* U (+ 1.0 (* (* J 2.0) (/ l U)))))

C

double code(double J, double l, double K, double U) {
	return U * (1.0 + ((J * 2.0) * (l / U)));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u * (1.0d0 + ((j * 2.0d0) * (l / u)))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U * (1.0 + ((J * 2.0) * (l / U)));
}

Python

def code(J, l, K, U):
	return U * (1.0 + ((J * 2.0) * (l / U)))

Julia

function code(J, l, K, U)
	return Float64(U * Float64(1.0 + Float64(Float64(J * 2.0) * Float64(l / U))))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U * (1.0 + ((J * 2.0) * (l / U)));
end

Wolfram

code[J_, l_, K_, U_] := N[(U * N[(1.0 + N[(N[(J * 2.0), $MachinePrecision] * N[(l / U), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 68.2%

   \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 68.2%

      \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot J\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. associate-*r* 68.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Simplified 68.2%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

6. Taylor expanded in U around inf 71.0%

   \[\leadsto \color{blue}{U \cdot \left(1 + 2 \cdot \frac{J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]
7. Step-by-step derivation

   1. associate-*r/ 71.0%

      \[\leadsto U \cdot \left(1 + \color{blue}{\frac{2 \cdot \left(J \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)\right)}{U}}\right) \]

   2. associate-*r* 71.0%

      \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(2 \cdot J\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}}{U}\right) \]

   3. *-commutative 71.0%

      \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(J \cdot 2\right)} \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right) \]

8. Simplified 71.0%

   \[\leadsto \color{blue}{U \cdot \left(1 + \frac{\left(J \cdot 2\right) \cdot \left(\ell \cdot \cos \left(0.5 \cdot K\right)\right)}{U}\right)} \]

9. Taylor expanded in K around 0 60.5%

   \[\leadsto U \cdot \left(1 + \color{blue}{2 \cdot \frac{J \cdot \ell}{U}}\right) \]
10. Step-by-step derivation

    1. associate-*r/ 60.5%

       \[\leadsto U \cdot \left(1 + \color{blue}{\frac{2 \cdot \left(J \cdot \ell\right)}{U}}\right) \]

    2. associate-*r* 60.5%

       \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(2 \cdot J\right) \cdot \ell}}{U}\right) \]

    3. *-commutative 60.5%

       \[\leadsto U \cdot \left(1 + \frac{\color{blue}{\left(J \cdot 2\right)} \cdot \ell}{U}\right) \]

    4. associate-/l* 62.0%

       \[\leadsto U \cdot \left(1 + \color{blue}{\left(J \cdot 2\right) \cdot \frac{\ell}{U}}\right) \]

    5. *-commutative 62.0%

       \[\leadsto U \cdot \left(1 + \color{blue}{\left(2 \cdot J\right)} \cdot \frac{\ell}{U}\right) \]

11. Simplified 62.0%

    \[\leadsto U \cdot \left(1 + \color{blue}{\left(2 \cdot J\right) \cdot \frac{\ell}{U}}\right) \]

12. Final simplification 62.0%

    \[\leadsto U \cdot \left(1 + \left(J \cdot 2\right) \cdot \frac{\ell}{U}\right) \]
13. Add Preprocessing

Alternative 13: 54.3% accurate, 44.6× speedup

Math

\[\begin{array}{l} \\ U + J \cdot \left(\ell \cdot 2\right) \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 (+ U (* J (* l 2.0))))

C

double code(double J, double l, double K, double U) {
	return U + (J * (l * 2.0));
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u + (j * (l * 2.0d0))
end function

Java

public static double code(double J, double l, double K, double U) {
	return U + (J * (l * 2.0));
}

Python

def code(J, l, K, U):
	return U + (J * (l * 2.0))

Julia

function code(J, l, K, U)
	return Float64(U + Float64(J * Float64(l * 2.0)))
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U + (J * (l * 2.0));
end

Wolfram

code[J_, l_, K_, U_] := N[(U + N[(J * N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 85.0%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

3. Taylor expanded in l around 0 68.2%

   \[\leadsto \color{blue}{\left(2 \cdot \left(J \cdot \ell\right)\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]
4. Step-by-step derivation

   1. *-commutative 68.2%

      \[\leadsto \left(2 \cdot \color{blue}{\left(\ell \cdot J\right)}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

   2. associate-*r* 68.2%

      \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Simplified 68.2%

   \[\leadsto \color{blue}{\left(\left(2 \cdot \ell\right) \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right) + U \]

6. Taylor expanded in K around 0 58.1%

   \[\leadsto \color{blue}{U + 2 \cdot \left(J \cdot \ell\right)} \]
7. Step-by-step derivation

   1. *-commutative 58.1%

      \[\leadsto U + \color{blue}{\left(J \cdot \ell\right) \cdot 2} \]

   2. associate-*r* 58.1%

      \[\leadsto U + \color{blue}{J \cdot \left(\ell \cdot 2\right)} \]

8. Simplified 58.1%

   \[\leadsto \color{blue}{U + J \cdot \left(\ell \cdot 2\right)} \]

9. Final simplification 58.1%

   \[\leadsto U + J \cdot \left(\ell \cdot 2\right) \]
10. Add Preprocessing

Alternative 14: 37.4% accurate, 312.0× speedup

Math

\[\begin{array}{l} \\ U \end{array} \]

FPCore

(FPCore (J l K U) :precision binary64 U)

C

double code(double J, double l, double K, double U) {
	return U;
}

Fortran

real(8) function code(j, l, k, u)
    real(8), intent (in) :: j
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = u
end function

Java

public static double code(double J, double l, double K, double U) {
	return U;
}

Python

def code(J, l, K, U):
	return U

Julia

function code(J, l, K, U)
	return U
end

MATLAB

function tmp = code(J, l, K, U)
	tmp = U;
end

Wolfram

code[J_, l_, K_, U_] := U

Derivation

1. Initial program 85.0%

   \[\left(J \cdot \left(e^{\ell} - e^{-\ell}\right)\right) \cdot \cos \left(\frac{K}{2}\right) + U \]
2. Add Preprocessing

4. Applied egg-rr 31.1%

   \[\leadsto \left(J \cdot \color{blue}{8}\right) \cdot \cos \left(\frac{K}{2}\right) + U \]

5. Taylor expanded in J around 0 43.6%

   \[\leadsto \color{blue}{U} \]

6. Final simplification 43.6%

   \[\leadsto U \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024115 
(FPCore (J l K U)
  :name "Maksimov and Kolovsky, Equation (4)"
  :precision binary64
  (+ (* (* J (- (exp l) (exp (- l)))) (cos (/ K 2.0))) U))